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Does V equal L?

Published online by Cambridge University Press:  12 March 2014

Penelope Maddy
Penelope Maddy*
Affiliation:
Department of Philosophy, Department of Mathematics, University of California at Irvine, Irvine, California 92717, E-mail: [email protected], E-mail: [email protected]
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Does V = L? Is the Axiom of Constructibility true? Most people with an opinion would answer no. But on what grounds? Despite the near unanimity with which V = L is declared false, the literature reveals no clear consensus on what counts as evidence against the hypothesis and no detailed analysis of why the facts of the sort cited constitute evidence one way or another. Unable to produce a well-developed argument one way or the other, some observers despair, retreating to unattractive fall-back positions, e.g., that the decision on whether or not V = L is a matter of personal aesthetics. I would prefer to avoid such conclusions, if possible. If we are to believe that L is not V, as so many would urge, then there ought to be good reasons for this belief, reasons that can be stated clearly and subjected to rational evaluation. Though no complete argument has been presented, the literature does contain a number of varied argument fragments, and it is worth asking whether some of these might be developed into a persuasive case.

One particularly simple approach would be to note that the existence of a measurable cardinal (MC) implies that V ≠ L,1 and to argue that there is a measurable cardinal. The drawback to this approach is that its implying V ≠ L cannot then be counted as evidence in favor of MC, as it often is. Indeed, there seems to have been considerable sentiment against V = L even before the proof of its negation from MC,2 and this sentiment must either be accounted for as reasonable or explained away as an aberration of some kind.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 1 , March 1993 , pp. 15 - 41
Copyright
Copyright © Association for Symbolic Logic 1993

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References

1 Scott, D., Measurable cardinals and constructible sets, Bulletin de l'Académie Polonaise des Sciences, vol. 7(1961), pp. 145–149Google Scholar.

2 See, for example, Scott's own reaction: ‘It did turn out that the existence of measurable cardinals was inconsistent with V = L, but so much the worse for the “unnatural” constructible sets!' (Quoted from the foreword to Bell, J. L., Boolean-Valued Models and Independence Proofs, Oxford University Press, Oxford, 1977, p. xii)Google Scholar.

3 For definitions, proofs, references, see, e.g., Drake, F., Set Theory, North-Holland, Amsterdam, 1974, Chapter 8Google Scholar.

4 I hope to discuss these issues in a sequel to this paper.

5 See Gödel, K., Collected Works, volume II (edited by Feferman, S.et al), Oxford University Press, Oxford, 1990, p. 185Google Scholar.

6 Moschovakis, Y., Descriptive Set Theory, North-Holland, Amsterdam, 1980, p. 610Google Scholar.

7 I would like especially to thank John Steel and Michiel van Lambalgen for their most helpful criticisms of earlier drafts and their generous suggestions for improvement, many of which have been adopted here (especially in §§3.2 and 3.3), and Mark Wilson for calling my attention to the historical sources exploited in §2 and for helping me to understand them to the extent that I do. (See his ‘Honorable intensions', to appear in S. Wagner and R. Warner, eds., Notes Against a Problem, for discussion of some related points.) I am also indebted to Jon Barwise, John Burgess, Keith Devlin, John Etchemendy, Ronald Jensen, Tony Martin, Colin McLarty, Gregory Moore, Yiannis Moschovakis, and Stewart Shapiro for conversations, comments and corrections. Finally, I am grateful to the NSF (DIR — 9004168) and to UC Irvine for their support, and to Ermanno Bencivenga and Kurt Norlin for help with translations. (It should not be assumed that any of these people agrees with the line I take here.)

8 Oxford University Press, Oxford, 1990.

9 Some of the weaknesses in the case for realism are considered in my A problem in the foundations of set theory. Journal of Philosophy, vol. 87 (1990), pp. 619–28Google Scholar; Taking naturalism seriously, to appear in the proceedings of the 9th International Congress of Logic, Methodology and Philosophy of Science, and Indispensability and practice, Journal of Philosophy, vol. 89 (1992), pp. 275–89Google Scholar.

10 Quine, W. V., Things and their place in theories. Theories and Things, Harvard University Press, Cambridge, MA, 1981, pp. 1–23Google Scholar. The quotation is from p. 21.

11 Einstein and Infeld describe these developments in chapter one of The Evolution of Physics, Simon and Schuster, New York, 1938. Interestingly enough, they call Mechanism a ‘philosophical' view, but their discussion of how such philosophical views are formed (‘generalizations… founded on scientific results') and how they function (‘they very often influence the further development of scientific thought by indicating one of the many possible lines of procedure') indicate that they operate within the realm of scientific practice, as understood in the previous section of the text. (The parenthetical quotations are from p. 51).

12 Einstein and Infeld, op. cit, pp. 152–3.

13 I include ‘mathematical' under the umbrella term ‘scientific'.

14 See Youschkevitch, A. P., The concept of function up to the middle of the 19th century. Archive for the History of the Exact Sciences, vol. 16 (1976), pp. 37–85Google Scholar. The following discussion of the history of the function concept owes a great deal to Youschkevitch's paper.

15 For this paragraph and the next, see also Bos, H. J. M., Differentials, higher-order differentials, and the derivative in Leihnizian calculus, Archive for the History of the Exact Sciences, vol. 14 (1974–1975), pp. 1–90CrossRefGoogle Scholar. Youschkevitch, op. cit., p. 55, suggests that Newton's notion of ‘fluent' involves a distinction between dependent and independent variables and goes beyond its usual geometric and kinematic interpretations, bringing it closer than Leibniz's notion to our modern idea of a function of a real variable.

16 Youschkevitch, op. cit., p. 57.

17 This quotation comes from another useful reference: Bottazzini, U., The Higher Calculus, Springer-Verlag, New York, 1986, p. 9Google Scholar.

18 Youschkevitch, op. cit., p. 61. At this time, Euler also allowed ‘mixed' functions, that is, piecewise analytically-definable functions.

19 Bottazzini, op. cit., p. 9.

20 Euler, unlike Descartes, did not intend his definition to exclude anything, but (as described in the next subsection) it was later adopted in just such a limitative spirit by the true Delinabilist d'Alembert.

21 My discussion of this episode draws on C. A. Truesdell, The Rational Mechanics of Flexible or Elastic Bodies, 1638–1788, his introduction to Euler, L., Opera Omnia, series 2, volumes 10 and 11, Turici, Leipzig, 1960Google Scholar. Some of the relevant writings of d'Alembert, Euler, and Daniel Bernoulli are excerpted in Struik, D. J., ed., A Sourcebook in Mathematics 1200-1800, Princeton University Press, Princeton, NJ, 1986, pp. 351–368Google Scholar.

22 See Truesdell, op. cit., p. 244, footnote 3. Cf. Bos, op. cit., pp. 56–7.

23 Truesdell, op. cit., p. 288.

24 Bottazzini, op. cit., p. 26.

25 Truesdell, op. cit., p. 246.

26 Truesdell, op. cit., p. 248.

27 Truesdell, op. cit., p. 276.

28 Truesdell, op. cit., p. 274.

29 Truesdell, op. cit., p. 285.

30 For the former, see Truesdell, op. cit., pp. 289–91, for the latter, Truesdell, op. cit., p. 286, Bottazzini, op. cit., p. 27, Youschkevitch, op. cit., p. 72, and, for example, Weinberger, H. F., A First Course on Partial Differential Equations, Blaisdell, New York, 1965, pp. 15–16Google Scholar. Truesdell writes, ‘… it is Euler's signal merit to have been led by a most secure intuition to results which the subsequent course of mathematics and rational mechanics has justified in all detail, though he himself lacked the experience and the apparatus to present an adequate argument for them' (p. 297).

31 Youschkevitch, op. cit., p. 70.

32 Youschkevitch (op. cit., §13) sees Euler's conception as the source of the modem one, while Truesdell (op. cit., p. 247, and ‘Introduction', in Euler, L., Opera Omnia, series 2, vol. 13, (1956), pp. ix–cvGoogle Scholar) holds that ‘[t]he possibility of a totally discontinuous function does not appear to have been regarded in the eighteenth century' (p. xlii). Bottazzini (op. cit., p. 217) believes that the modern conception only emerged with the work of Riemann (see §2.3 below).

33 Truesdell, in the introduction cited in the previous note, p. xlii.

34 See Truesdell, , Rational Mechanics, pp. 261, 277–8, 281–2, 284–5, 297–9Google ScholarPubMed.

35 Truesdell, op. cit., p. 282. For this section, in addition to Bottazzini, op. cit., and Youschkevitch, op. cit., I rely on Hawkins, T., Lebesgue's Theory of Integration, second edition, Chelsea, New York, 1970, especially Chapters 1 and 2Google Scholar.

36 Bottazzini, op. cit., p. 72.

37 Bottazzini, op. cit., p. 73.

38 Bottazzini, op. cit., p. 77.

39 See Hawkins, op. cit., pp. 9–12.

40 Bottazzini, op. cit., p. 242. See also Hawkins, op. cit., p. 17.

41 Bottazzini, op. cit., pp. 215–6.

42 In fact, Dirichlet's celebrated general definition, though stressing the arbitrariness of the correspondence, was restricted to continuous functions. See Bottazzini, op. cit., pp. 196–201, and Youschkevitch, op. cit., pp. 78–9.

43 Op. cit., p. 217.

44 See Hawkins, op. cit., Chapter 2.

45 Hawkins, op. cit., p. 29.

46 This, and Darboux's previous remark, are quoted by Hawkins, op. cit., p. 27.

47 Hawkins, op. cit., p. 122. For this section, my main sources are Monna, A. F., ‘The concept of function in the 19th and 20th centuries', Archive for the History of the Exact Sciences, vol. 9 (1972) pp. 57–84Google Scholar, and Moore, G. H., Zermelo's Axiom of Choice, Springer-Verlag, New York, 1982Google Scholar, especially §§1.7 and 2.3 and Appendix 1.

48 Monna, op. cit., p. 69. As the translation is mine, and I am no linguist, I include the French: ‘Il y a fonction, dès qu'on image [‘imagine' in the original French] une correspondance entre des nombres, qu'on convient de considérer comme les états de grandeur d'une même variable y, avec d'autres nombres, tous distincts, qu'on convient de considérer comme les états de grandeur d'une même variable x. On ne s'occupe pas, dans cette définition, de rechercher par quels moyens la correspondance peut étre effectivement établie; on ne cherche même pas s'il est possible de l'établir. La notion de fonction, entendue de cette manière, est entièrement contenue dans la notion de détermination; ce point de vue s'oppose à celui qui consiste à partir de certaines fonctions simples, et à considérer des expressions composées avec ces fonctions simples, en réservant le mot de fonction aux expressions ainsi obtenues.'

49 See Baire, R., Lecons sur les fonctions discontinués, Gauthier-Villars, Paris, 1905, p. viGoogle Scholar.

50 Actually, the Baire classes are defined to be disjoint; each class consists solely of the new functions obtained by the specified means.

51 Moore, op. cit., p. 68.

52 Monna, op. cit., p. 71. ‘Bien que, depuis Dirichlet et Riemann, on s'accorde généralement à dire qu'il y a fonction quand il y a correspondance… sans se préoccuper de procédé qui sert à établir cette correspondance, beaucoup de mathématiciens semblent ne considérer comme de vrais fonctions que celles qui sont établies par des correspondances analytiques. On peut penser qu'on introduit peut-être ainsi une restriction assez arbitraire…' And below, ‘Ainsi il n'est pas évident qu'il existe des fonctions non représentables analytiquement; il y a donc lieu de rechercher s'il existe de telles fonctions…'

53 He cannot. See Moore, op. cit., p. 99.

54 Moore, op. cit., §§1.7 and 2.3. Moore writes, ‘In the wake of Zermelo's proof, all of these mathematicians faced a painful reappraisal of their researchers, one which some of them carried out more fully, openly, and consistently than others' (p. 76).

55 Moore, op. cit., p. 314.

56 Moore, op. cit., p. 318.

57 It was these remarks of Hadamard that got me started on this line of thought in the first place.

58 Monna, op. cit., pp. 74–5. ‘Les résultats acquis, dès la fin du XIXe siècle, ont surabondamment prouvé combien était simpliste l'opinion d'après laquelle it serait possible de limiter le champ des Mathématiques à l'étude d'une catégorie déterminée de fonctions … le but définitif de ces recherches pathologiques doit être la délimitation des fonctions considérées comme saines.'

59 Moore, op. cit., p. 100. This is Moore's paraphrase of Lebesgue.

60 Their views become more radical over the years, as described in Moore, op. cit., pp. 92–103.

61 See Moore, op. cit., p. 313.

62 Reference in Moore, op. cit., p. 92.

63 Moore, op. cit., gives a wonderfully readable account of this history.

64 Bernays, P., ‘On platonism in mathematics', in Benacerraf, P. and Putnam, H., eds., Philosophy of Mathematics, second edition, Cambridge University Press, Cambridge, 1983, pp. 258–271Google Scholar. The quotation is from pp. 259–260. My term ‘Combinatorialism' is not present in Bernays, whose actual terminology I will sketch in a later note.

65 Bernays, op. cit., pp. 263–4.

66 Bernays, op. cit., pp. 258–9, 260.

67 Bernays calls the anticonstructivist attitude ‘platonism' (op. cit., p. 259). The acceptance of the completed infinite and the ‘quasicombinatorial' understanding of sets, sequences and functions, are both ‘platonistically-inspired mathematical conceptions' (pp. 259–60). I use the term ‘Combinatorialism' to emphasize the quasicombinatorialist aspect of Bernays thinking and to avoid the term ‘platonism' which has so many other meanings in the philosophy of mathematics.

68 Bernays, op. cit., p. 261. Bernays actually uses his own term: ‘platonism'. See previous note.

69 V = L becomes actually limitative only with the arrival of Scott's theorem.

70 Devlin, K., The Axiom of Constructihility, Lecture Notes in Mathematics, vol. 617, Springer-Verlag, Berlin, 1977, pp. 27–28Google Scholar.

71 Quoted in Wang, H., From Mathematics to Philosophy, Routledge and Kegan Paul, London, 1974, p. 10Google Scholar. See also Gödel's, Collected Works, vol. II, p. 136Google Scholar.

72 Collected Works, vol. II, p. 152Google Scholar.

73 Gödel himself is not fully persuaded by this argument. He leaves open the question of whether or not all ordinals can be legitimately considered definable.

74 Collected Works, op. cit., p. 152.

75 Ibid.